Flags

We’d like to have matrix-oriented versions of Engel’s theorem and Lie’s theorem, and to do that we’ll need flags. I’ve actually referred to flags long, long ago, but we’d better go through them now.

In its simplest form, a flag is simply a strictly-increasing sequence of subspaces of a given finite-dimensional vector space. And we almost always say that a flag starts with and ends with . In the middle we have some other subspaces, each one strictly including the one below it. We say that a flag is “complete” if — and thus — and for our current purposes all flags will be complete unless otherwise mentioned.

The useful thing about flags is that they’re a little more general and “geometric” than ordered bases. Indeed, given an ordered basis we have a flag on : define to be the span of . As a partial converse, given any (complete) flag we can come up with a not-at-all-unique basis: at each step let be the preimage in of some nonzero vector in the one-dimensional space .

We say that an endomorphism of “stabilizes” a flag if it sends each back into itself. In fact, we saw something like this in the proof of Lie’s theorem: we build a complete flag on the subspace , building the subspace up one basis element at a time, and then showed that each stabilized that flag. More generally, we say a collection of endomorphisms stabilizes a flag if all the endomorphisms in the collection do.

So, what do Lie’s and Engel’s theorems tell us about flags? Well, Lie’s theorem tells us that if is solvable then it stabilizes some flag in . Equivalently, there is some basis with respect to which the matrices of all elements of are upper-triangular. In other words, is isomorphic to some subalgebra of . We see that not only is solvable, it is in a sense the archetypal solvable Lie algebra.

The proof is straightforward: Lie’s theorem tells us that has a common eigenvector . We let this span the one-dimensional subspace and consider the action of on the quotient . Since we know that the image of in will again be solvable, we get a common eigenvector . Choosing a pre-image with we get our second basis vector. We can continue like this, building up a basis of such that at each step we can write for all and some .

For nilpotent , the same is true — of course, nilpotent Lie algebras are automatically solvable — but Engel’s theorem tells us more: the functional $\lambda$ must be zero, and the diagonal entries of the above matrices are all zero. We conclude that any nilpotent is isomorphic to some subalgebra of . That is, not only is nilpotent, it is the archetype of all nilpotent Lie algebras in just the same way as is the archetypal solvable Lie algebra.

More generally, if is any solvable (nilpotent) Lie algebra and is any finite-dimensional representation of , then we know that the image is a solvable (nilpotent) linear Lie algebra acting on , and thus it must stabilize some flag of . As a particular example, consider the adjoint action ; a subspace of invariant under the adjoint action of is just the same thing as an ideal of , so we find that there must be some chain of ideals:

where . Given such a chain, we can of course find a basis of with respect to which the matrices of the adjoint action are all in ().

In either case, we find that is nilpotent. Indeed, if is already nilpotent this is trivial. But if is merely solvable, we see that the matrices of the commutators for lie in

But since is a homomorphism, this is the matrix of acting on , and obviously its action on the subalgebra is nilpotent as well. Thus each element of is ad-nilpotent, and Engel’s theorem then tells us that is a nilpotent Lie algebra.